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Theorem 0npi 6468
Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
Assertion
Ref Expression
0npi ¬ ∅ ∈ N

Proof of Theorem 0npi
StepHypRef Expression
1 eqid 2056 . 2 ∅ = ∅
2 elni 6463 . . . 4 (∅ ∈ N ↔ (∅ ∈ ω ∧ ∅ ≠ ∅))
32simprbi 264 . . 3 (∅ ∈ N → ∅ ≠ ∅)
43necon2bi 2275 . 2 (∅ = ∅ → ¬ ∅ ∈ N)
51, 4ax-mp 7 1 ¬ ∅ ∈ N
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1259  wcel 1409  wne 2220  c0 3251  ωcom 4340  Ncnpi 6427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2947  df-sn 3408  df-ni 6459
This theorem is referenced by:  elni2  6469
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